Group 7 Nisarg Shah : Siddharth Dhakad : Chirag Sethi : Shiv Shankar : Dileep Kini : Self Referencing and Gödel's Incompleteness Theorem

Overview Self Reference Self Referencial Statements and Paradoxes Attempts at Resolution Truth and Provability Completeness, Soundness and Consistency Gödel's Incompleteness Theorem Douglas Hofstadter's Proof Typographical Number Theory (TNT) Tarski's Undefinability Theorem Some Implications of Gödel's Theorem Gödel vs Artificial Intelligence References

Self Reference - Any object that refers to itself or its own referent Quines in computing Recursion in programming Self referential statements in linguistics Courtesy : Courtesy :

Tupper's Self Referential Formula The points which satisfy the above inequality when plotted form the following graph: Courtesy :

Self Referential Statements and Paradoxes Liar’s Paradox "This sentence is false.“ “The next sentence is false. The previous sentence is true." Epimenides Paradox (7th century BC) Epimenides (who himself was a Cretan) said : "All Cretans are liars”

Russell's Paradox for Sets S = the set of all sets that are not members of themselves Is S a member of itself? If S is an element of S, then S is a member of itself and should not be in S. If S is not an element of S, then S is not a member of itself, and should be in S.

Attempts at Resolution Tarski's hierarchy of languages o introduce the constraint that elements of a given level may only refer to elements of lower levels Prior's assumption o every statement includes an implicit assertion of its own truth. o S => "This sentence is true and S" Three valued Logic o Dialetheism - Graham Priest o True, False, Both true and false o Identifies Liar's sentence as both true and false o Inherent problems

Truth and Provability T-schema by Alfred Tarski to assign truth value of a statement by inductive definition o Alfred Tarski, "The Concept of Truth in Formalized Languages", Corcoran, J. ed. Logic, Semantics and Metamathematics, 1983 Semantic theory of truth Provability on the other hand is being able to be derive a sentence using language's axioms and inference rules

Completeness and Consistency Completeness of a formal system: o each true formula of the language of the system must be derivable in the system o Truth => Provability Consistency of a formal system: o for no formula A of the language of the system both A or ¬A are theorems of the system Godel's Incompleteness Theorem: In any consistent system capable of expressing elementary arithmetic, provability is a weaker notion of truth.

Gödel's Incompleteness Theorem

Kurt Gödel Kurt Gödel, born April 28, 1906, was an Austrian - American logician, mathematician and philosopher. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century. Gödel is best known for his two incompleteness theorems, published in 1931 when he was 25 years of age, one year after finishing his doctorate at the University of Vienna.

Historical Perspective In the nineteenth and early twentieth centuries, one of the big mathematical goals was to reduce all of number theory to a formal axiomatic system. Russell and Whitehead's Principia Mathematica was the most famous attempt Gödel's theorem dashed this hope completely Gödel showed that for any formal axiomatic system capable of expressing elementary arithmetic, there is always a statement about natural numbers which is true, but which cannot be proven in the system The Gödel sentence combines three elements: the representation of provability, self-reference and negation.

Douglas Hofstadter's Proof Hofstadter, Douglas R., "Gödel, Escher, Bach: An Eternal Golden Braid”, Proof by Contradiction Defines a formal axiomatic system called Typographical Number Theory (TNT) Assumes TNT to be complete and consistent, encapsulates all of mathematics perfectly Leads to contradiction

Typographical Number Theory (TNT) TNT is an attempt to represent all of math in an axiomatic way. TNT expresses everything in terms of a few simple symbols. o Mathematical Symbols : +, *, = o Variables : a, a', a'', a''',... o Logical Symbols : A, E, v,^, ~ o Numbers : 0, S0, SS0, SSS0,...

TNT Statements o S0 + SS0 = SSS0 Axioms o Aa:~Sa=0 o Aa:(a+0)=a o Aa:Aa':(a+Sa')=S(a+a') o Aa:(a*0)=0 o Aa:Aa':(a*Sa')=((a*a')+a) Rules o ~~ can be removed from any statement o Au:~ and ~Eu: are interchangeable anywhere inside a string. o... (Refer Appendix A for all rules)

TNT TNT-Theorem: TNT-Statement which can be inferred from TNT-Axioms and TNT-Rules Example: S0+S0=SS0 Aa:Aa':(a+a')=(a'+a) (Commutativity) ~Ea:a*a=SS0 (No root of 2) ~(Ea':Ea'':SSa'*SSa'' = SSSSS0) (Primality of 5) Can even express exponent or in general any statement about numbers Assumption: Any true statement can be derived in TNT

Gödel's Statement in TNT G: This statement is not a theorem of TNT If G is false, then it is a theorem of TNT. Then we have a valid theorem which is false, this is not possible. Hence G is true and it is not a theorem of TNT Hence there is a true statement in TNT which is not provable in TNT proving the theorem. Question: TNT has statements about numbers. How do we write G in TNT?

Gödel Numbering Replace every symbol with three digit unique number a S = and so on... Every TNT-Statement thus has a unique Gödel number Rules become functions on natural numbers [Kurt Gödel, "On formally undecidable propositions of Principia Mathematica and related systems", Monatshefte für Mathematik und Physik, 1931]

Gödel Numbering Every TNT-Statement thus has a unique Gödel number o Example ~ E a : a * a = SS Rules become functions on natural numbers o Example Dummy Rule: Whenever a string ends in the symbol "000", you can replace that symbol with "005". The same fake rule, written differently: Whenever a number is a multiple of 1000, you can add 5 to it.

Theoremhood Theoremhood of a number: Being derivable by applying certain functions (Rules) on certain initial numbers (Axioms) Theoremhood thus becomes a mathematical function and hence can be represented by TNT by our assumption that TNT can express all of mathematics Benefit of Gödel Numbering: A statement can talk about a statement of TNT by using its Gödel number Difficulty: Gödel number of a statement is longer than the statement itself => Intelligent way of referring

Arithmoquining Mechanism for writing a TNT sentence about another TNT sentence Take a sentence with one free variable and replace all occurrences of the free variable with the Gödel number of the sentence. Example: T: a=S0 A: Sentence T is 1 where “Sentence T” is the Gödel number of T Arithmoquining is thus a mathematical function taking as input Gödel number and giving Gödel number of its arithmoquine and can be represented in TNT.

Finally G is here!!! T: The arithmoquine of a is not a valid TNT theorem- number G: The arithmoquine of Sentence T is not a valid TNT theorem-number G uses "arithmoquine of Sentence T" to intelligently refer to itself G in words: "This statement is not a valid TNT theorem" G is true and not a theorem of TNT as shown before

Some Implications of Gödel’s Theorem No consistent system of axioms whose theorems can be listed by a computer program is capable of proving all facts about the natural numbers No sufficiently strong (and useful) reasoning system can derive all truths Gödel's second incompleteness theorem (assumption of consistency within the system leads to inconsistency) Hilbert's second problem cannot be solved within arithmetic

Notion of truth inside Formal system The statement of the Gödel's theorem invokes the notion of truth. There is a precise characterization of what it means for a statement in a formal language to be true within a given structure for that language This characterization is independent of any syntactic considerations, and in the case of formalized arithmetic it is not expressible in the theory itself (unlike the notion of theoremhood) - John W Dawson Jr

Tarski's Undefinability Theorem The theorem says that given some formal arithmetic, the concept of truth in that arithmetic is not definable using the expressive means of that arithmetic It is possible to define a formula True(x) whose extension is set of all true statements in the theory, only by drawing on a meta-language whose expressive power goes beyond that of L, say second-order arithmetic. No sufficiently powerful language is strongly-semantically- self-representational This implies a major limitation on the scope of "self- representation"

Gödel vs Artificial Intelligence Gödel's theorem has been taken to imply that we humans have an innate ability to recognize a valid proof when we see one. "Mathematical thinking (and hence conscious thinking generally) is something that cannot be encapsulated within any purely computational model of thought" - Penrose Reason is creative and original. Although any bit of reasoning can be codified into a set of rules, there will always be further exercises of reason.

Gödel vs Artificial Intelligence Human mathematician cannot be accurately represented by an algorithmic automaton. For any such automaton, there would be some mathematical formula which it could not prove, but which the human mathematician could both see, and show, to be true. o Minds, Machines and Gödel - J. R. Lucas This argument is based on the inherent assumption that human mind can escape Gödel's result !!! Guarantee the last line of Lucas's argument !!!

Gödel vs Artificial Intelligence Gödel's theorem shows that if human reasoning was computable, then it had to either be unsound, or it had to be impossible for human to know both what human's own reasoning powers were and to also know that they were sound Gödel's theorem places limits on all 'sufficiently' expressive reasoning systems (including human mind) This Gödelian limitation is not due to lack of human intelligence, but is inherent in any reasoning system that is capable of reasoning about itself -McCulloug,"Can Humans Escape Gödel?"

Godel vs Artificial Intelligence One can never entirely understand oneself, since the mind, can be sure of what it knows about itself only by relying on what it knows about itself All the limitative theorems of mathematics suggest that once the ability to represent your own structure has reached a certain critical point it guarantees that you can never represent yourself totally. - Hofstader, "Godel,Escher, Bach" So when we ask "Is Artificial Intelligence possible?" Gödel does not rule out AI.

Gödel vs AI As a concluding remark to the above argument regarding Godel's theorem and mind we quote an ancient philosopher and poet :- But as for certain truth, no man has known it, Nor shall he know it, neither of the gods Nor yet of all the things of which I speak. For even if by chance he were to utter The final truth, he would himself not know it: Xenophanes

References Kurt Gödel, "On formally undecidable propositions of Principia Mathematica and related systems", Monatshefte für Mathematik und Physik, Hofstadter, Douglas R., "Gödel, Escher, Bach: An Eternal Golden Braid”, Alfred Tarski, "The Concept of Truth in Formalized Languages", Corcoran, J. ed. Logic, Semantics and Metamathematics, 1983 J. R. Lucas, "Minds, Machines and Gödel", Oxford Philosophical Society, 1959

References D. McCullough, "Can Humans Escape Gödel?", ms theorem.html

Thank You!

Appendix A : Rules of TNT All the rules for the Propositional Calculus are also rules of TNT Rule of Specification: If ∀ u: x{u} is a theorem, then so is x{u} and so is x{v/u} where v is any term Restriction: v cannot be quantified within x{u}. If it's true for all, it's true for one. Rule of Generalization: If x{u} is a theorem, then so is ∀ u: x{u} Restriction: u must be free If something is true for something without restrictions, then it's true for all. Rule of Interchange: The strings ∀ u:~ and ~ ∃ u: are interchangeable If it isn't true for all, then there doesn't exist any for which it's true.

Appendix A : Rules of TNT Rule of Existence: Any instance of a term may be replaced by u, if ∃ u: is placed in front. Restriction: The term cannot contain variables that aren't free. Rules of Equality: Symmetry: If r = s is a theorem, then so is s = r Transitivity: If r = s and s = t are theorems, then so is s = t Rule of Successorship: Add S: If r = t is a theorem, then so is Sr = St Drop S: If Sr = St is a theorem, then so is r = t Rule of Induction: If ∀ u: is a theorem and x{0/u}> is a theorem then ∀ u: x{u} is a theorem Restrictions: x{u} is well-formed; u is free